Quasi-interpolators with application to postprocessing in finite element methods

TL;DR

This work develops quasi-interpolators that map into conforming piecewise polynomials of degree $p+1$ using local weight functions of degree $p$, achieving an optimal $\mathcal{O}(h^{p+2})$ approximation when the target is sufficiently close to the $L^2$ projection onto degree-$p$ polynomials. A second family using piecewise-constant weights enables effective postprocessing for HDG methods, while both families admit applications to projection operators in negative-order Sobolev spaces. The authors provide rigorous construction (vanishing-trace and non-boundary) and existence results for the weight functions, along with detailed stability and approximation bounds, and demonstrate the practicality via numerical experiments on mixed FEM, DPG, and HDG. This framework offers a conforming, derivative-free postprocessing alternative with broad potential in improving FEM accuracy and in analyzing negative-order operators. The work also identifies open questions in extending the weight-function existence to three dimensions and in refining patch choices for constants-based constructions.

Abstract

We design quasi-interpolation operators based on piecewise polynomial weight functions of degree less than or equal to $p$ that map into the space of continuous piecewise polynomials of degree less than or equal to $p+1$. We show that the operators have optimal approximation properties, i.e., of order $p+2$. This can be exploited to enhance the accuracy of finite element approximations provided that they are sufficiently close to the orthogonal projection of the exact solution on the space of piecewise polynomials of degree less than or equal to $p$. Such a condition is met by various numerical schemes, e.g., mixed finite element methods and discontinuous Petrov--Galerkin methods. Contrary to well-established postprocessing techniques which also require this or a similar closeness property, our proposed method delivers a conforming postprocessed solution that does not rely on discrete approximations of derivatives nor local versions of the underlying PDE. In addition, we introduce a second family of quasi-interpolation operators that are based on piecewise constant weight functions, which can be used, e.g., to postprocess solutions of hybridizable discontinuous Galerkin methods. Another application of our proposed operators is the definition of projection operators bounded in Sobolev spaces with negative indices. Numerical examples demonstrate the effectiveness of our approach.

Figures (6)

• Figure 1: Left triangulation does not satisfy Assumption \ref{['ass:vertices']}. Right triangulation satisfies Assumption \ref{['ass:vertices']} with $R=4$.
• Figure 2: Visualization of patches.
• Figure 3: Errors $\|u-J_{0}^{p+1} u\|_{\Omega}$ (left) and $\|u-I_{0}^{p+1} u\|_{\Omega}$ (right) with $u(x,y)=\sin(\pi x)\sin(\pi y)$ and domain $\Omega = (0,1)^2$.
• Figure 4: Errors $\|u-I_{0}^{p+1} u\|_{\Omega}$ with $u(x,y)=\sin(\pi x)\sin(\pi y)$ and domain $\Omega = (0,1)^2$ for $p=2,3$ when using different patch sizes for the definition of the weight functions. For the larger patch a $(p+1)$-order patch is used, for the smaller one a $p$-order patch.
• Figure 5: Errors of approximation and postprocessed solution for the mixed FEM described in Section \ref{['sec:numeric:mixed']}.

Theorems & Definitions (23)

• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof
• Theorem 5
• Theorem 6
• Remark 7
• Lemma 9